Rectifiable metric spaces: Local structure and regularity of the Hausdorff measure. (English) Zbl 0806.28004

The author considers \(n\)-rectifiable metric spaces \((X,\rho)\). He shows analogously to the Euclidean setting: For almost all \(x\in X\) with respect to the \(n\)-dimensional Hausdorff measure \({\mathcal H}^ n_ \rho\) (understood as in Federer’s book) there exist a norm \(\|\cdot\|_ x\) on \(\mathbb{R}^ n\), a map \(\phi_ x: X\to \mathbb{R}^ n\), and a closed set \(A_ x\subset X\) such that \(\phi_ x(x)= 0\), \[ \lim_{r\to 0} (\alpha(n) r^ n)^{-1} {\mathcal H}^ n_ \rho(B_ \rho(x,r)\cap A_ x)= 1, \] and \[ \limsup_{r\to 0} \left\{\left|1- {\|\phi_ x(y)- \phi_ x(z)\|_ x\over \rho(y,z)}\right|;\;y,z\in A_ x\cap B_ \rho(x,r),\;y\neq z\right\}= 0. \] He gives also an extension of ideas due to Kolmogoroff concerning maximal and minimal strongly metrically invariant measures. He has found an integral representation of these measures on the space \(C([0,1])\). As a consequence, the equivalence class of Banach-Mazur compactum containing \(\|\cdot\|_ x\) consists of all norms on \(\mathbb{R}^ n\) induced by an inner product provided that the maximal and minimal Kolmogoroff measures coincide.


28A78 Hausdorff and packing measures
26B10 Implicit function theorems, Jacobians, transformations with several variables
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